On the Hardy space H 1 o f a C 1 d o m a i n

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On the Hardy space H 1 o f a C 1 d o m a i n

Eugene B. Fabes and Carlos E. Kenig

Introduction

In this work we will extend the notion of the classical Hardy spaces H p, p ~ 1 of the unit disc, or of the half space, R~_=R"-I• oo), to the case of a bounded C 1 domain D of R". We first recall the definitions of these spaces (see [18]): In the case of R~_, HP(R~_)= {3= (ul, ..., u,), ~ satisfies the generalized Cauchy--Riemann equations, i.e.

O u j _ aui , Ouj Oxi Oxj ' Y~j=a Oxj - O,

and (0)*, the non-tangential maximal function of~, is in LP(R"-a)}. To such a system we associate the function f = u , lR.-l, which is in LP(R"-J). In every simply connected domain, the conditions on t] are equivalent with ~ = V U , AU=O on R~_, a n d f t h e n becomes --~-n" It is well known that, for p > 1, the mapping ~ f OU induces an isomorphism onto LP(R"-]), and ~ may be recovered f r o m f a s the vector formed by the Poisson integral of the Riesz transforms o f f and the Poisson integral o f f . Another way of describing this procedure is the following: for XER~_, let S f ( X ) =

e. fR--1

iX_y[,_ 2 1 f ( y ) d y , where for definiteness we have taken n=>3. Then, O(X)=VSf(X). In the case p = 1, we no longer get all of LI(R"-]), but only a subspace, called HI(R"-1). This space has been extensively studied (see [18], [9], [10]). It was proved in [18] that f E H I ( R "-a) i f f f a n d its Riesz transforms R , f b e l o n g to L 1 (R"-a). Under these conditions, 17 is recovered from f as before. It was shown in [9] that H I ( R " - 1 ) * = B M O , the space of functions of bounded mean oscillation introduced by John and Nirenberg. Also, C. Fefferman observed that as a consequence of this duality, fC H 1 (R"- 1) iff f = ~ , 2j a j, where ~ ' ]2jl < 0% supp aj c Bj, Bj abaU, IlajliL=<=,--j-7,, 1

faj=O.

This was later on extended to p < l b y R . R. Coif-

I~,jl

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man ([2]) in the case of

H p

of the real line, and for the n-dimensional case by R. Lat- ter ([16]).

It is this type o f definition and properties that we wish to extend to the case o f bounded C 1 domains. We consider only the case l<-p<oo. Our definition o f

HP(D)

is the following:

HP(D)={~=VU, AU=O,

such that

(~)ELP(OD)}.*

When D is simply connected, this corresponds precisely to solutions, a, of the generalized C a u c h y - - R i e m a n n system. T o any such 0

=VU,

we associate the function f = 0U on 0D. f i s in

Lg(OD)={f~LP(OD),

f ~ o f = 0 } . In the case l < p < ~ o , it follows from the results in [7] that this induces an isomorphism onto L~ (0D), and that can be recovered from f by

~=VS(Tf)

( , ) , where T is an operator bounded and invertible on

L~(OD).

We study the case p = 1 in this paper, obtaining results anal- ogous to the ones described for the flat case of R~_: we give a characterization in terms of Riesz transforms, an atomic decomposition, and a duality pairing with BMO (0D). Moreover, we show that if

fEHI(OD),

then we can recover ~ from it by ( , ) .

In the case when D =

{zEC,

]z[< 1}, our spaces also essentially coincide with the classical H a r d y spaces. To be specific, classically (see [3]), Re

HI'(OD)-= {fELI'(OD)

such that

f=uloD,

where

u+iv=F

is analytic in

D, FELI'(OD)}.*

It is not difficult to see then that using our definition o f

HP(OD), H~'(OD)=ReHP(OD)n

{mean value 0}.

This remark also generalizes to the unit ball B, in R". F o r this particular example the case p > 1 of our results was established by Koranyi--V~gi ([15]), and the case p = 1 has recently been studied by Ricci and Weiss ([17]), who obtained the atomic decomposition and the singular integral characterization. O f course, these authors relied on specific formulas and properties available for the case o f B, but unavailable in the general case. T o substitute these we use the theorem of A. P. Calder6n ([1]) together with the results and techniques o f [7], extended to the end point cases o f p = 1 and BMO, and an extension o f a result o f Varopoulos ([20]).

Before beginning the major part o f this work we need to introduce some of the basic notations and definitions we will use,

Capital letters, X, Y, Z, will denote points of a fixed domain D c R " . Lower case letters x, y, z are reserved for points in R "-1. The notation (X, Z ) denotes the inner product in R" whereas x - z will be used for the inner product in R "-a.

Points on the boundary o f

D, OD,

will usually be denoted by Q and sometimes by P. Also letters t, s will be reserved for real numbers.

Definition.

A domain D ~ R " is called a C 1 domain if corresponding to each point

QCOD

there exists a ball, B, with center Q and a coordinate system o f R"

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On the Hardy space H ~ of a C 1 domain

with Q as the origin such that with respect to these new coordinates

Bc~D

= B n {(x, t): x E R "-1, t > r

I~)~Cl(Rn-1),

0 ~

r = 0 = -b-Z, (0), i = l, ...,

.)}

and

B 0D= B {(x,

We will assume t h r o u g h o u t this p a p e r that b o t h D and R " \ D are connected.

I f D is a b o u n d e d C 1 domain we will let NQ denote the unit inner normal to

ODatQ. Given

0 < e < l we set

F~,,~(Q) = {XE D: IX-QI < ,5, (X-Q, NQ) > cclX-QI}

and

/~,,o(O) = { x E a " \ / ~ :

IX-al < ,~, ( X - Q , N e)

< - c ~ l X - a I } .

In general when the numbers a and 5 are understood we will drop them as sub- scripts and write

F(Q)

and /~(Q) respectively for the interior and exterior cone with vertex Q.

By a surface ball with center

QEOD

and radius R > 0 we mean the intersec- tion o f OD with a ball in R" of radius r and center Q. We will use the notation

S,(Q)

for such a surface ball. Since our domain D is bounded, it is obvious that there exists a constant A such that, for any

QEOD,

and

r~A, Sr/2(Q)=OD.

Hence, we will restrict our attention to surface balls of diameter less than or equal to A.

F o r these balls, there exist constants cl and c2, depending only on D such that

Clr("-l)<=a(S~r(Q))<=e2r ("-~).

Here,

a(E), EcOD

denotes the surface area o f the set E.

I f we consider the distance on

OD

inherited f r o m R", the balls for this distance coincide with our surface balls, and the triple

(OD, d, a)

becomes a space of homogeneous type (see [3]).

We now introduce the spaces we will be mainly concerned with:

Definition.

B M O ( 0 D ) =

{fEL2(OD)

for which there is a constant c such that for all surface balls S~

1

where

fsr = ~ fs, f(Q) dQ.

We let tlfllBMOOm=inf {c: ( . ) holds for c}. Hence, if we identify two functions differing by a constant, B M O

(OD)

becomes a Banach space. It is well-known ([3] page 593) that an equivalent n o r m is obtained on B M O

(OD)

if we replace the L~-means in ( . ) by LP-means, l < _ - q < ~ .

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If fE BMO (0D), we define II fll bmo ~OD)= 11 f]l BMO (OD) -~- II f[I L~(~D). With this definition, bmo (OD) is a Banach space.

Definition.

A real valued function a defined on OD is called an atom if there exists a surface ball Sr such that s u p p a c S r ,

fooa(Q)dQ=O,

and

1

IlallL~<ov) ~-~fSr ) 9

9 ( , ) f = ~ i = 1 2 i a i , where the

ai's

Definition. h~(OD)={fEL~(OD),

such that * -

are atoms, and ~ = ~ 12i]<~o}. If

fEhI(OD), Ilf[lhl(oo)=inf~=~

IAi], such that (*) holds.

It is well known (see [3]) that with this norm,

h~(t)D)

becomes a Banach space, and

(h~(OD))=SMO(OD).*

Here, the duality pairing is given by

foof. adQ,

where f E B M O (gD) and a is an atom.

In the first section we study the continuity and compactness on BMO (~D) of the integral operator obtained by restricting to the gD the classical double layer potential.

Precisely we consider the principal value operator

where

1 (P-Q' No)f(Q) dQ,

Kf(P)

= ~ p.v.

f OD IP-QI"

a~, = area of {XER", I S l -- 1}.

(n => 2)

We then apply these results to the study of the Dirichlet problem with boundary data in BMO (OD) and the N e u m a n n problem with boundary data in h I(0D).

In the second section the results on the N e u m a n n problem with boundary data in h 1 (OD) are used to study the H a r d y space H 1 (OD). The main result of this paper, Theorem 2.6, is proved in this section.

We remark that the Dirichlet problem with boundary data in BMO (OD), on a bounded starshaped Lipschitz domain has been previously considered in [8]. What is important to us here is that for C ~ domains the solution can be expressed in terms o f the double layer potential.

Section 1.

Theorem 1.1. K: BMO (OD)~BMO (OD),

and is in fact compact on this space.

Proof.

Our first remark is that K i s well defined on BMO (OD) since K ( c ) = 1 c for any constant c. (See [7], page 170.)

Fix f E B M O ( O D ) and a surface ball

Sr(--Sr(Po))

of radius r and center

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On the Hardy space H x of a C ~ domain

PoCOD.

N o w ,

1 ( P - Q ,

NQ)

Kf(e)=-d~fs~, IP-Ql" (f(Q)-fs,,)dQ

1

+ - - L~ ('-Q" IP-QI" <,o-Q,

I P o - Q I "

1 ( e o - Q, NQ) d, 1

where

A(r, Po)=-~foo\s~" JPo-QI" (f(Q)-fs~.) Q+-~fs~,.

Using the fact that K is continuous on

L2(OD),

it is easy to see that

1

f v/=

(a~S~) f s. ]Kf(P)- A (r, Po)l' dP) :,,<- c (a---~,) s,. ]f(Q)-fs=']=)

n--12 f (P-O, Nq) (Po-Q, Nq) dP If(Q)-fs.J dQ + ^?,

a ao\s=. . IP-QI"

1 P 0 - a l "

[[If[l~

f o r lf(O) Q]

r MO(0D)"~ D\s=. I P 0 - Q I "

-fs,.Id <- clfI[BMO0O).

To show the compactness o f K on B M O it will suffice to show compactness on bmo. By the use o f a finite partition o f unity we m a y further reduce the problem to the compactness on b m o o f the operator,

f~K($f),

where

$ECo(Bn)

and Bn is a ball o f radius 6 > 0 and center on

OD

such that

B~ n O D = {(x, # ( x ) ) : x E R "-~,

~C co~({Ixl

< 1}),

IV~l-<-

m0}

(see [7]). Choose now 0E C o (B n) such that 0 - 1 on a neighborhood o f the support of ~. It is easily seen that (1

--O)K$

is c o m p a c t on bmo. Our p r o b l e m is reduced to the study o f the compactness o f

OK$.

We now pick a sequence

(%}~Co({Ixl<l})

such that ~ b ~ 0 and V ~ y - ~ V ~ uniformly on R "-a and for f ~ b m o we define

f(P) = 0 (P) f oo kj (P, Q) ~ (Q)f(Q) dQ

where kj is defined on

Ban n ODXB4n n OD

as

k i(?, Q) -. r 1

[ i x - zl=+(%(x)- %(z))=] ":= r + IW(z)l =

(P = (x, ~b (x)) and Q = (z, ~b (z)).)

The operator Kj is c o m p a c t on bmo.

We now show that

K:-,-OIf$

on b m o

(019).

The first observation we need is that

Kj-,-OKt~

on

Lt'(OD),

for all 1 < p < oo.

N o w let

g(P) = ~, (P) .f(P).

Then [1

g[lbmo (ao) <=

^C

11

fllbmo (OD), and g is supported on

ODnB n.

Define now a function ~ on R "-1, by

~(x)=g(x,~(x)).

Then

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~EBMO (R(n-~)), [[ffl[~Mo(a.-~)-<_e Ilg[lbmo(Oo), and supp f f c {lxl< 1}. We now introduce a sequence of operators on R"-X:

and

1 ~ (x) - 9 (x) - w ( z ) . ( x - z)

xCg(~) = W. p.v.

f ({x_zl~+lV(x)_r g(z)d~,

1 ~ j (x) + # i ( z ) - v # j ( z ) . (x - z)

K ~ j ~ ( x ) = ~ 2 " p.v.

f (ix_zl2+l%(x)_%(z)l~).,~ ~(z)dz.

It is easy to verify that for any constant

e, K~(e)=K%(e)=O.

Moreover, if

PEB4o c~

OD, P=(x,

~(x));

O(P)K(~f)(P)=O(P).K~(~)(x),

and

_Ky(P)=O(P).K%(~)(x).

As K%~_K~ in LP(R "-1) l < p < ~ , , ~-+~b and V4~j~V4~ uniformly, using the proof of the first part of the theorem, we see that ]](K%--K,)~I/SMO(R._I)= <

ejJ]g]laMO(R--1), where

ej~j~=O.

Using these facts once more, it follows that if

S,(Po)

is a surface ball on

OD,

such that

Sr(Po)cB4~,

then

1

f s.(Po) [0 (P) (K~f) (P) - Kj f(P) -

[0K(0f)

- Kj f]s.[ dP

(s.

(P0))

O"

~j(I}gll,MO(R"-~)-k II~)]L--~(R--~)) =<

eej

IIf[]bmo(ao).

Now, as

Kj~OK~b

in

LP(OD)

for l < p < o o and from the above estimate in BMO

(OD),

we conclude that

Ki~OK ~

in bmo

(OD).

Theorem 1.2.

~ I+ K is invertible on

BMO

(OD), and a -~ I - K . is invertible on kl(OD). (K denotes the adjoint of K.)*

Proof

It was shown in [7] that

-}I+K

is invertible on

L2(OD).

(It was actually shown in [7] that

-~I+K

is invertible on

L2(OD)

for n ~ 3 . The result however remains true in dimension n = 2 and the proof given in [7] is valid also for this case.

Referring the reader to the proof of Theorem 2.1 in [7] one only needs to observe that i f f satisfies ( ~ - I + K * ) f = 0 then

foo fdQ=-O

and, therefore

foolog[X-Q[f(Q)dQ

= o ( [ x [ -1) as (Ix[ -~o~).

From the invertibility on

L ~ (OD)

it is immediate that ~ I + K is 1 - 1 on BMO

(OD),

and, hence, by Theorem 1.1 it is invertible on BMO

(OD).

It was also shown in [7] that -~

I - K *

is invertible on the space

= Lo I = 0}.

L~(OD)

was stated in [7] only for n=>3.

(Again the invertibility of

TI--K*

on

The proof given there (Theorem 2.5 in [7]) is also valid for n = 2 since, as pointed out above,

foologlX-QIf(Q)dQ~O

as I x l ~ provided

foof=O.)

^{The in-}

1 L~(OD)

implies the same for the adjoint operator, ~

I - K ,

vertibility of T

I - K *

on

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O n the H a r d y space H 1 o f a C 1 d o m a i n

on the quotient space, L2(OD)/constants. In particular I I - K is 1 - 1 on BMO (OD), and so, by Theorem 1.1, it is invertible.

Since the dual space of hl(OD) is BMO (see [3]) the invertibility of - ~ I - K ~ * on h~(OD) will follow from the invertibility of I I - K on BMO provided we know that y I - K * 1 is indeed continuous on hl(OD). The validity of the continuity on h~(OD) is in fact a consequence of the continuity on h 1 of a space of homogeneous type of a general class of operators discussed by Coifman and Weiss on pages 598-- 600 in [3].

The invertibility of -~1 I - K * on h~(OD) immediately suggests the solvability of the Neumann problem for the Laplace operator in the form of a single layer potential. In the next few results we want to formalize the notion of solvability.

Theorem 1.3. Suppose a is an atom. Let

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Sa(X) = c, D [X---Z-~ "-2 dQ c, - (n--2)o~, for n >= 3 and

Sa(X) = foDlOglX-Qla(Q)dQ for n = 2.

Given 0 < a < 1, there exists 6~, o such that

(VS,)*(Q) = sup [VxSQ(X)[, QEOD

FQ

belongs to L 1 (OD) and

faD (VSa)* dQ <= c, independent of a.

(Recall FQ={XED: [X-QI<5~ and (X-Q,N~2)>o~IX-Q[}.)

Proof. Assume a is supported in the surface ball, St, of radius r > 0 . Now f a dQ=O and [[a[IL=(Oo)<=er 1-" with e depending only on OD.

L s , (7S,)* dQ <= cr "-1/2 (L~, (VS")*2 dQ) '1~<= cr"-l/2[[a[]L'(aD) <= c.

Let O denote the center of S,.

s . ( x ) = (k(X,

O_)-k(X, O))a(O)aO

1 x - O where k(X, Q) -

~o. l x - O l "

Set F p = { X ~ D : I X - P [ < & ( X - P , N e ) > a I X - P I } . Clearly for 5 small enough, depending only on a and D, rp n OD = {P} VPEOD. For

X ~ F e , lVxS,(X)I <-- c , [ X - Ol-" f%,~ I Q - 0 [ [a(Q)l dQ <= c, r l X - O [ - " .

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Eugene B. Fabes and Carlos E. Kenig Using this last estimate we have

f o\s,r

**(vso)*(p)**

dP <= c~.

As an immediate consequence we have the

Corollary. Given O < a < l , there exist 6 > 0 and fEhl(OD)

I](VS:)*IILI(oo) <- C[/ft[hlOo).

Here, as in Theorem 1.3,

s:(x) = f iX_Qt~ f(Q)dQ

1

S:(X) = f oo log ( X - Q ) f ( Q ) dQ and

e > 0 , such that for any

for n ~- 3, for n - - 2 (VS:)*(Q) --- Sup [VxS:(X)l.

FQ

We will now show that the Neumann problem is solvable in the form of a single layer potential with a density in h ~ (OD) provided o f course the data also belongs to h 1 (OD).

Theorem 1.4. Given gEhl(OD) there exists a unique (modulo constants) har- monic function, u(X), such that

i) for any 0 < a < l the function

(Vu)* (Q) = sup IVxu (X)[

FQ

belongs to LI(OD) and II(Vu)*[[L~(OD)~CflgIIL~ with C independent of g, and

ii) (Vxu(X), NQ)-~g(Q) pointwise for almost every Q~OD as XEFQ tends to O.

Moreover u may be written as the single layer potential of ( l I - K * ) - J g i.e.

u(X) : (x)

Proof. The invertibility of -~I-K1 * on h x (0D) (Theorem 1.2) implies it is suffi- cient for a p r o o f of Theorem t.4 to show the nontangential convergence almost everywhere o f the normal derivative o f

u(X) = s:(x)

to the value ( - ~ I - K * ) f when fCh~(OD). More precisely we would like to show that ( V x S : ( X ) , N Q ) ~ ( - ~ I - K * ) f ( Q ) f o r almost every Q~OD.

We write f = ~ = x 2 i a i where at is an atom and [ 2 d < ~ . From the Corol- lary o f Theorem 1.3

(vsz= ~,o,)*

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On the Hardy space H 1 of a C 1 domain

has arbitrarily small L 1 norm over

OD

for N sufficiently large. Hence the existence almost everywhere of the above limit and its equality with ( ~ - I - K * ) f will follow provided for almost every Q

(Vx Sa(X), NQ) > ~ (-~ I--K) a (Q)*

as

X-~Q, XEFQ,

when a is an atom. Since atoms in particular belong to L2(0D) this last fact was already shown in [7].

For the uniqueness part of the theorem, we need two lemmas. The first one, although not explicitly stated there in the form we need it, was proved in [7]. We first need some notation.

Since D is a C 1 domain, there exist 5 > 0 , and a finite covering o f the set

{X, dist(X, OD)~_5}

by balls

Bj=B(Pj, rj)

with center

PjEOD

such that

B(Pj, 4rj) c~ D=B(Pj, 4rj) n

{(x, y); y > ~j(X)}. (See the Introduction.)

Now, let {~j} be a finite partition o f unity for the set {X, dist

(X, OD)<=5},

subordinate to the cover {Bj}. We assume each

r

F o r each t > 0 and sufficiently small, we set

Dt,j = B(Pj,4r~) c~

{(x, y); x E R "-1, y > # j ( x ) + t } , and

Ft, j : B(Pj, 4rj) t~

{(x,

qbj(x)--}-t); xE

R"-I}.

Lemma 1.5.

Suppose that Au=O in D, and that for some p,

l < p < o o ,

c, (o,

where C is independent o f t and j, for sufficiently small t. Then, if u(X)~O as X-+

QCOD nontangentially for a.e. Q, u-~O in D.

The p r o o f is given in Theorem 2.3 o f [7].

L e m m a l . 6 .

Suppose Au=O in D, and for some

0 < a < l ,

(Vu)(Q)-~*

supr Q

]Vu(X)IELI(OD). I f - ~ Q =--O on OO, then u is constant in D. Ou

Proof.

Using the fundamental theorem o f calculus to express u in terms o f

Vu,

and the convexity o f cones, it is not hard to show that

uCL~(OD).*

Thus, u has a nontangential limit a.e. on

OD.

Now, pick a j, and a sufficiently small t. F o r

xER "-1,

let

fj, t(x):~tj(x,t+q~j(x)).u(x,t+(~j(x)).

Then, fj, t is in LI(R n-l) independently o f t, and Vfj, t(x) is in L~(R "-1) independently of t. Hence, by the Sobolev embedding theorem,

fj, tCL"-I/'-2(R"-a).

Thus, the nontangential limit of u on

OD

belongs to

L"-I/"-~(OD),

and u satisfies the hypothesis of Lemma 1.5, with

p = n - 1 / n - 2 .

Hence, if we show that

U]oo=e,

Lemma 1.6 will follow from L e m m a 1.5.

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10 Eugene B. Fabes and Carlos E. Kenig

We are going to show that

fa. u(O). ~(Q)dO=O

for every

~CC(OD)

such that

fao ~dQ=O.

This will certainly imply that

ulao=c.

Pick such a ~. Let B be the solution to the N e u m a n n problem

AB=O,

in D, ~ 0B ( Q ) = ~ ( Q ) on 0D (see [7]). As B is the single layer potential o f a function in

LP(OD)

V 1 < p < ~ , B is bounded on D. Moreover, it was shown in [7] that

(VB) (Q)~LP(OD)*

V 1 < p < oo.

Now, let

uloo = g "fop

g(O)" ~ ( O )

dO = .~y f oo OJ(O)" g(Q)" ~(Q) dQ.

f a. Ojg.r dQ = lim f ,. ~,j(Q,)u(Q,)(VB(Q,), NQ,)dQ,

= lim

f , (VOpju)(X), VB(X))dX.

t ~ O + t , j

f o, , (v(~ju), w ) , ~ x

⁼

f , , (Vu(Qt),

NQ,)Oj(Qt)B(Q,)dQ,

+ f,. (voj (Q,), NQ,? u (Qt) B (Qt) aQt - fot, j (A ~9j) (X). u (X) B (X) dX

- 2

f o , . (vo~ (x), vu (x)) B(x) ax.

Letting t-~O +, we have

-

O~j

faD ~hj. g' eb. dQ = J;o ~ (Q)" g (Q)" r (Q) dQ

- f o (A ~9j) (X). u ( X ) . B (X) d X - 2 f o

(V~/j

(X), Vu (X)). B (X) dX.

Adding in j, we get

Jot, g" ~" dQ = - f . A ( 2 j Oj)" u(x). B(X)dX

- 2 f~, (v Z j Oj(x), W,(X))B(X) dX

⁼

-f~,

^A

[(2J O j)" u].

^{B ( X )}

dX.

Let n o w ~ h = l - Z j f f j , then,

--A [(~__,jtkj).u]=A(~h.u).

Moreover, if

v=~hu,

v vanishes on a neighborhood o f

OD,

and so, an integration by parts shows that

f . A (v).BdX=O.

Hence,

fad g" ~'dQ=O,

and the l e m m a and also the uniqueness part o f T h e o r e m 1.4 are established.

We now turn to the Dirichlet p r o b l e m with B M O data. As mentioned in the introduction, this problem has already been treated in the more general case of Lipschitz domains in [8]. We will sketch an alternative approach for the case of C 1 domains.

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O n t h e H a r d y s p a c e H ~ o f a C a d o m a i n 11

Definition 1.7. A measure /t on D is called a Carleson measure if # {XED, [X--Q[<r}<=Mr "-1 for all QEOD. The least such M is called the Carleson norm of/~.

Theorem 1.8. Given gEBMO (0D), there exists a harmonic function, u(X) such that

i) The measure d # = d ( X ) . IVu(X)12dX is a Carleson measure on D, with Carleson norm bounded by a constant times the square of the BMO OD) norm og g;

here d ( X ) = d i s t (X, 0D).

ii) u ( X ) ~ g ( Q ) for almost every QEOD as XEFQ tends to Q. Moreover, u maybe written as

( . ) u(X) = ( X - Q , (ff

I+K)_I(g)(Q)dQ"

I X - Q l

iii) Conversely, if dl~=d(X). IVu(X)12dX is a Carleson measure, then there exists a function gEBMO (OD) such that u-*g nontangentially and such that (*) holds.

Proof. F o r the p r o o f of iii), we refer to [8]. Since gEL2(OD), using the uniqueness results o f [7], and Theorem 1.2, we see that all we have to show is that if u(X) =

( X - Q, NQ)

e, foo IX-Q]" f(Q)dQ, where f E B M O (OD), then /z as defined in (i) is a Carleson measure.

Fix a QoEOD, and an r > 0 . Let Sr-=Sr(Qo). Then f=(f-fs~r)Xs2,+

( f - f s ~ ) X s ~ + f s , = f l + f ~ + f s , r , and so u(X)=ul(X)+uz(X)+fs,,, and hence, Vu(X)=Vul(X)+Vuz(X). We note that

f

^{[AI ~ ~}IIflI~MO(OD)a(Szr), and so, f IZfll z <= CIIfll~MoOD)a(S2,) where T = ( - } I + K ) . Hence, if we use the fact that if u is harmonic, then

f d(X).lVu(X)J" f o. I.(Q)1"dQ,

(see [6]); the term corresponding to u~ is taken care of.

By the formula we have for u~, it is easy to see that

1 1

IVuz(S)l ~ c -offer ]Q0-Q]" I f ( Q ) - f s , r l dQ <= c .of~ ]Qo-Q]" I f ( Q ) - f s , r l dQ

c

< ^{~ - -} IlfllBMo(0 ^D).

Y

From this, the desired estimate for u~ easily follows.

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12 Eugene B. Fabes and Carlos E. Kenig Section 2.

In this section we will study Hardy spaces on D and we will identify them with spaces o f functions on OD. We start with three technical lemmas. The first one is by now classical:

( n - 2 ) Lemma 2.1. Suppose U is harmonic in D c R " . Let ff=VU. Then for P>=7-SS~"

Ia[v is subharmonic in D.

F o r a p r o o f o f this theorem, see [19], page 234.

Lemma 2.2. Suppose v(X) is continuous, nonnegath~e and subharmonic in D. In addition, assume there is a p, l < p ~ , such that for each or, 0 < ~ < I , v*ELV(OD), and assume v(X)~g(Q) as X-+Q nontangentially for almost every QEOD.

Let P(g)(X) denote the Poisson integral of g(i.e., u ( X ) - P ( g ) ( X ) satisfies Au=O i n D, u*ELV(OD) for each 0 < c ~ < l , and u o g non-tangentially almost everywhere). Then v(X)<=P(g)(X) in D.

Proof. There exists a sequence, {Vi(X)}, o f nonnegative subharmonic functions with each VjECZ(D) and such that V s ~ V uniformly on each compact sub- domain of D. In particular AVj>=O in D.

Take now a function ~,(X)ECo(D ) with ~ 0 in D, ~b,--1 on {XEB: dist (X, 0D) =>- e}

and 4 ~ 0 on

{XED: dist (X, 3D) <= el2}.

If G(X, Y) denotes the Green's function for the domain D (see [7, page 183]) we have 9 ~ (X) V,. (X) = - f o G(X, Y) A (Vj ~ ) (Y) dY

_<- 2

(v, o(x, r), v (r))vj(r)ar+f q(x,

Letting j-~ ~ we obtain the same inequality for V. It easily follows that the same inequality holds for the subharmonic function

W(X) = V ( X ) - e ( g ) ( X ) .

Since W*ELP(OD) (for each 0 < ~ < 1 ) and W(X)~O nontangentially at almost every point o f the boundary, the argument given in [7, page 184] shows that

lim f n (Vr G(X' Y)' Vrff)~(Y))W(Y)dY = O,

~ 0

and

lira f D G(X, Y ) A ~ ( Y ) W ( Y ) dY = O.

e~O

This of course implies W ~ 0 in D.

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On the Hardy space H ~ of a C a domain 13 We also need the following extension o f a result of N. Varopoulos ([20]), for the half-space, to the case of a bounded C ~ domain.

Lemma 2.3.

Assume

f ~ B M O

(OD). Then, there exists F(X)~CI(D) such that for almost every Q~OD, F(X)-+f(Q) as X ~ Q nontangentially, and

IVF(X)[

is density for a Carleson measure, with Carleson norm bounded by the

B M O

norm o f f Moreover, if we use the notation introduced before Lemma

1.5,

F = Z j Fj, where each Fj is compactly supported in B~,

[VF~.[

is the density Jbr a Carleson measure, and there exists gj~LV(OD) for all 1 <=p<o~ such that for t sufficiently small and

x = (x, + ; ( x ) + t ) < r , , j , IC(X)I ~

gj(x, +j(x)).

Proof

We m a y assume

foDf=O,

and hence

HfHbmoOD)~CHf[IBMOOD).

Let 0 j be as in L e m m a 1.5. The function

f , flx)=Oj(x , rbg(x)).f(x, cbj(x))

belongs to B M O ( R " - I ) . Fix now a non-negative function

KCC~~

with integral 1. By T h e o r e m 4.3 of [12], there exists a Carleson measure #io~ on R+

s u c h t h a t

fo/x)= f fa.+ Kr(x-z).dl~j.q,s(Z, y),

where

Kr(z)= y-("-l)K[v ).

More- over, the Carleson n o r m for

PrOs

is b o u n d e d by the B M O n o r m

o f f , j.

Fix now a C = function b on [0, ~ ) such that b ( t ) = l for 0 < = t ~ l / 2 , b ( t ) = 0 for t=>l. Let now

Fq, flx, t)=ffa,+Kj(x-z)b[@ld,~iofz, y ).

Then, arguing as in the second p r o o f of T h e o r e m 1.1 in [20], we see that

IVFoj(x , t)l

is a Carleson measure in R + , with Carleson constant bounded by the B M O n o r m o f f % . Moreover,

Foj(x, t) ~t-o

fcj(x), and IF0flx,

t)l<= f fR,+ K,(x-z)ldpIo (z, y)l--gj(x), gj(x)

is in B M O (R n-l) and, hence, locally in every

L p

space, 1 ~ p < ~o.

Pick now

OjCCo(Bj), Oj--1

in a neighborhood of the support o f ipy. Set now, for

X6Bjc~D, X=(x,y),

y > ~ j ( x ) ,

Fj(X)=Oj(X).Foj(x,y--~y(x)) ,

and

Fj~O

outside

Bj c~ D.

Let now F ( X ) = ~ j

Fy(X).

Certainly,

F(X)~CI(D),

and for almost every

QCOD, F(X)-~f(Q)

nontangentially. Moreover, if

X~Bjc~D, X=(x,~j(x)+t)CFt, j,

then [Fj(X)I_-<

Io&, t+%(x))l.lro~(x, t)[<-lo~(x,t+Oj(x))[ [gj(x)[<--e.)~W~oD(X , q)j(X)).g~(x).

All that remains to show then, is that VFj(X) is a Carleson measure on D with Carleson n o r m b o u n d e d by the B M O n o r m o f f It is enough to consider balls

B,=B(Qo )

with 0<_-2r~A, where A is as in the introduction,

QoEOD.

As we can assume that if

X~Bj~D, X=(x, ~j(x)+y),

then dist (X,

OD)~-y,

we see that if

B,c~Bjc~D#O,

and

XCB, c~Bjc~D,

then there exists a constant c depending only on D such that

Br ~ Bj c~ DcB~((x,

~j(x))) c~ D. Thus, it is enough to check that for the function

[Vx, t(Oi(x, t+~j(x)).Foflx, t))[

and for any ball

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14 Eugene B. Fabes and Carlos E. Kenig Aa contained in R "-a, with radius

6<=cA,

the

f2 f ~ ]v~,,(Oj(x, t+~j(x)). Fojx, t) I dx dt <-- c. 3(n-1)I[fI[BMO(OD).

Now,

s IF+,(x, t)l <tx dt ~ f ~ L o leotx, ')-SoXx)I <tx d,+a f~o ls'o,(x)lax.

a fa~ IA,(x)l dx <= 6"-* Ilf0,llL(--l>~.-,)

<=

can--lllfllbmoOD) ~ Ca"--*

[IfIIBMo(aD).

Also,

s

<

^{: . C}

.6". lifo II

j BMOORn-1) ~ c * A .~n-1

IIf[I

BMO(OD)*

As we already know the required estimate for

VxaF, flx, t),

the p r o o f of the lemma is completed.

Definition 2.4.

(a) HV(D)= { R = ( U l , . . . , t/n) ; such that g = V U , U harmonic in D, and for some 0 < e < l the function

Iffl*(O)=suprQ l~(x)]

belongs to

LP(OD)}.

Here we will consider only the case 1 <=p< ~.

Remark.

If D is simply connected, the vectors a = V U ,

AU=O

on D coincide with the set o f vectors a = ( u , ..., u,) such that A ~ = 0 , d i v a = 0 , and

Ou~ Ouj

Oxj Oxk'

i.e. g satisfies the generalized C a u c h y - - R i e m a n n equations, and our definition coincides with the one used by Stein and Weiss in [18].

F o r a vector ~ in

HP(D),

we set

II~ll/~,w>= II I~]*[IL.(aV)"

F r o m the results o f H u n t - - W h e e d e n [11], and Dahlberg [4], we know the existence o f non-tangential limits for each

~HP(D)

at almost every (surface measure) point

Q~OD.

In particular, if 17 = V U, then

lim

(Ne, VU(X)) = OU

x - a

-ff-N-~e (Q )

non-tang.

exists for almost every Q. M o r e o v e r ,

-~-~-]CLP(OD), OU

and a localization procedure

. O U 0

as the one used in Lemma 1.6 shows that

Jab-ff-~e = 9

Definition 2.4.

(b)

H'(OD)={fCLV(OD); f(Q)=(No,~(Q)),

for ~TCH'(D)}.

We also set II

fll~l,(OD)=ll I~]IIL,oo).*

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On the Hardy space H ~ of a C ~ domain 15 Remark. I f l < p < ~ , the results in [7] s h o w t h a t HP(OD) with the n o r m defined a b o v e is a B a n a c h space which coincides as a s e t with LPo(OD)=

{f~LP(OD); fool=O} m o r e o v e r the H ~ n o r m is e q u i v a l e n t to the L p n o r m . Hence, we will restrict ourselves f r o m n o w on to the case p = 1. L e m m a 1.6 shows t h a t H 1 (OD) is a B a n a c h space, w h i c h can be identified with H 1 (D).

L e m m a 2.5. Assume gE HI(D). Then

II~ll~lco~lll~lllL~0o~-Hence,

if f 6 Hl(OD), f ( O ) = (No,/t (O)), II

fll~lOD)

~- 111~l

IILI(0D).

Proof. U s e L e r n m a 2 . 1 to pick r, 0 < r < l such t h a t v = ] g ] ' is s u b h a r m o n i c in D. By L e m m a 2.2, with p = 1/r, we see t h a t

v~e(lal').

A s ]v*[a/r<=[e(lff[')*]ll', we see t h a t

L o ["]* dQ <- L o [P([fi[')*]l/~ dQ <- c L o 1"1 dQ, a n d hence the L e m m a is established.

Remark. This L e m m a also shows t h a t different ct's yield c o m p a r a b l e n o r m s in H ~ (D) and in H ~ (0D).

W e are n o w r e a d y to state o u r m a i n t h e o r e m , which gives several equivalent c h a r a c t e r i z a t i o n s of/-/1

(0D).

T h e o r e m 2.6. Assume that fE LI(~D), f oD f=O. Then, the following contritions are equivalent:

i) f ( H I ( O D ) , i.e. f ( Q ) - ~ ( N e , ~ ( Q ) ) , fi = VU, AU = 0, and (~)*EL~(OD).

ii) f6hl(OD), i.e. f =

~=lJ, lai,

where the al are atoms, and ~'1211 < + c o . iii) (VSy)*E Z 1 (OD).

f0 (t,j- Q;) ....

iv) R j f ( P ) = c.p.v. D ~ Jt~d)dQ, 1 = j = n, belong to < "< LI(0D).

Moreover, all the corresponding norms are equiva&nt, and if f satisfies any of the (1 I K * ] - x f

equivalent conditions, ~-~ -- j is defined, and ~=VS((I/2) r r , ) _ l , .

A s a consequence o f this t h e o r e m , a n d its p r o o f , a n d o f T h e o r e m 1.8, we obtain the following result for B M O (0D):

Theorem 2.7. The following conditions are equivalent for a function f in L ~ (OD):

i) f is in B M O (OD).

ii) The measure d(X)IVu(X)I~dX on D is a Carleson measure, here d ( X ) = dist (X, OD), and u is the solution to the Diriehlet problem with f as boundary data.

iii) f is the boundary value of an F6CI(D), such that IV F 1 is a Carleson measure, and F satisfies the conditions in Lemma 2.3.

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iv) f = f o + ~ . = a R j f j , where f o , f l . . . . , f , EL=(OD), and Rj are the operators defined in Theorem 2.6.

We now turn to the p r o o f of Theorem 2.6. This will be accomplished in several stages. We will first show that i)+-~ii).

Theorem 1.4 shows ii)~i). The p r o o f that i ) ~ i i ) is accomplished in two steps.

In the first one, we show the equivalence for all starshaped C 1 domains, and in the second step we pass to the general case. We now study the starshaped case.

Lemma 2.8. I f D is a starshaped C ~ domain, the continuous functions with mean value zero are dense in HI(OD). In particular, for starshaped C ~ domains, h~(OD) is dense in H 1 (OD).

Proof. We may assume that D is starshaped with respect to the origin. F o r a given ~EH~(D), set ~t(X)=~(tX), 0 < t < l . The function (NQ,~(tQ))=ft(Q) is continuous, with mean value zero, hence, it belongs to h~(OD). Moreover, by Lemma 2.5,

][ (No, ~(tQ) - O(Q))[IHI(OD) =

f oD

1~ (tQ) - O(Q)[ dQ.

This last integral converges to zero as t converges to 1, since I~]*ELI(OD).

Theorem 2.9. Given gE BMO (0D), define for each fE H 1 (OD) lg ( f ) --- f , {VG(x), VU(x)) dX,

where G is the function constructed in Lemma 2.3, and ~=VUEHI(OD) satisfies - - - OU ( Q ) = / ( Q ) . Then

ONe

[tg (f)[ <_- c l] gIIBMO(OD) "

][f]IHIOD),

where c depends only on D. Moreover, if for p > 1, fE H I(OD) c~ L p (OD) = LPo (OD), then lg ( f ) - - fad g .fdQ.

Furthermore, if D is starshaped, and IEHI(1)D) *, then there exists a unique gEBMO (OD) such that l=lg.

Proof. By L e m m a 2.1, we can find a number r, 0 < r < l , such that ]VU(X)[ r is subharmonic in D. By L e m m a 2.2, IVU(X)]<-P[IVU[']I/'(X) in D. Hence,

fD IVa(X)l IVU(X)I dX

^{< -}

f . Iva(x)[P(IVUIr)I/r(X)dX,

and since ]VGI is the density for a Carleson measure, we have the last integral bounded by e. [I g][ BMO (~D) f~D IV U(Q)[dQ <= e. U gll BMO (Oo)" II f]l/~I(~D), (see [5]).

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On the Hardy space H 1 of a C 1 domain 17

Remark.

The above p r o o f can also be obtained using a purely geometric argument, without using the subharmonicity property o f the gradient, or the results in [5].

Assume now that

fEL~(D),

l < p < co. Then, by [7],

(VU)ELV(OD).*

lg ( f ) =

f o {VG,

VU)

dx = 2 j f o

{VGj, VU)

dx;

here we are using the notation in L e m m a 2.3. Moreover,

f D (VGj,

VU)

dx = f Bja, (VGj,

VU)

dx

: tlim

f D,,j (VGj,

VU)

dx,

since the integral on the left hand side is absolutely convergent by the first part o f the proof. Now, using Green's theorem, we see that

f Dt, s (VGj,

VU)

dx = f rt, j Gj(Q,) . (VU(Qt), Not > dQt.

Moreover, since

[Gj(Qt)I<=gj(Q), gjELP'(OD),

and

(VU)~L'(OD),*

the dominated convergence theorem shows that the last integral converges to

fop Gj(Q).f(Q)dQ

as t ~ 0 . Adding now in j, we see that lg

( f ) = f o D g . f . d Q .

Assume now that D is starshaped, and

IEH~(OD) .*

Since

hl(OD)=HI(OD),

there exists gCBMO

(OD)

such that for any atom,

a, l(a)=foDa(Q)g(Q)dQ=

0t}o

fD(VU a, VG(X))dX,

where

Ua(X )

is harmonic in D, and satisfies - ~ - ~ - = a , and

G(X)

is the function constructed in Lemma 2.3. Using the first part of the present theorem, and Lemma 2.8, we have

l ( f ) = f D (VU, VG)dx

for all

fEHI(OD),

with

AU=O

on D, and - ~ o = f

OU

The uniqueness o f g is obvious.

Remark.

Once we establish the equivalence of i) and ii) for general C 1 domains the condition that D be starshaped in the last part o f Theorem 2.9 can be dropped.

Also, Theorem 2.7 is then seen to follow from Theorem 2.6 and Theorem 2.9, using the argument given on page 145 of [10].

Corollary 2.10.

I f D is a starshaped C 1 domain, then h~(OD)=HX(OD).

Proof.

By L e m m a 2.8,

hl(OD)

is dense in

HI(OD).

By Theorem 2.9, it is closed in

HI(OD).

Hence, both spaces are equal. We now show that i)+~ii) for arbitrary C ~-domains.

Theorem 2.11.

For D a bounded Cl-domain, hl(OD)=HI(OD).

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Pro@

We just have to show

HI(OD)chl(OD).

Let

fEHI(OD);

let a E H I ( D ) be such that ~ = V U, f = - ~ a

OU

on 0D.

Using a regularized version (to make it of class C 1) of the construction given in Section 2 of [21], we see that for each point

QEOD,

we can find a ball

B=Br,

with center Q and radius r, and we can construct a Cl-starshaped domain DB such that

B4rnDDD,,ODn~B2, nOD,

and with the additional property that

E H ~ (D)

^c

H ~ (On).

We then cover 0D by a finite number,

{B j},

o f the above balls, and we let

~kiEC~(0D ) be a finite partition o f unity of

OD

subordinate to the above cover.

F r o m Corollary 2.10, fj=(f~, NDBj) belongs to

h~(ODB).

Since we may assume atoms are supported in surface balls o f diameter smaller than any preassigned positive number, we can write

~kjf=~jgj,

where

gjEh~(OD).

We now observe that if

hEhl(OD),

and

~EC~(OD),

then

~,.h-m(~k.h)Eh~(OD),

here m ( ~ h ) =

a(OD)

1

fooO.h.

^Since

foof=O,

~ j m ( O i / ) = O , and so ~ j m ( O j g j ) = O . More- over, f = z ~ j

~ljf=Xj ~ljgj=Zj (0igi-m(~kigi))=~J h~,

where

hjEhl(OD),

and thus

fEhl(OD).

Remark.

Theorem 2.11 shows that as sets h I ( 0 D ) = H I(0D). However, since h 1 is continuously embedded in H 1, the open mapping theorem gives the equivalence o f the two norms.

We proceed with our p r o o f of Theorem 2.6.

The corollary to Theorem 1.3, shows that ii)~iii). F o r the converse implication, we need the following uniqueness result:

Lemma2.12.

Assume fELI(OD); f oof=O. Assume that llVxSf(X)llnlw)=O.

Then,

f = 0 .

Proof.

L e m m a 1.6 implies that

Sr

a constant for all

XED.

In R " \ D , Sr is harmonic, and since

foDf=O, Ss(X)=O(IXt 1-")

as [ X [ ~ . Also it is easily seen that the function ~f*(Q)=supr Q [Sr (/~e is the truncated exterior cone defined in the introduction.) belongs to

LP(OD)

for some p > l , and moreover, limx~Q, xerQ

Sr

exists for almost every

QEcgD.

This exterior nontangential limit is the same as the interior one, which is, of course, the constant e.

Let us assume that e # 0 , a n d for simplicity we take e > 0 . Choose a ball B so large that D c B and 1Sr162 on

OB,

with r an arbitrary but fixed positive number.

F r o m Lemma 2.2, we have

]Sr VXEB\D,

and from this we con- elude

ISf(X) l-<e

in all o f R " \ D . Hence, either S I ( X ) = _+_ c in all of R " \ D

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O n the H a r d y space H x of a C x d o m a i n 19

or

tSs(X)l<c

in all of R " \ D . Hence, at almost every point

Q~OD,

0 ~_ lim

(NQ, VSI(X)) = (-~ I+K)f(a).*

X~f2 XEPQ

Moreover, from the interior normal derivative, we have 1

T f - K

* f = 0 on 0D, and hence

K f*

has mean value 0 on OD. We then have (~

l+K)f=O,*

and so f = 0 on

OD.

I f the constant e equals 0 we easily deduce that

S~(X)

is zero also in R'~,xD, and once again, the jump relations on the normal derivative give f = 0 on 0D.

We now prove iii)~ii). Assume f satisfies iii). By the equivalence of i) and ii), we have that

( ~ I - K ) f = a - ~* Ss~hl(cgD).

By Theorem 1.2, there exists

fEhl(OD)

such that t,I~Q

(--i Z , K * ) f = (~ Z - K W .

The function

Scr_I)(X)

satisfies the hypothesis of Lemma 1.6, and hence it is iden- tically constant in D. By Lemma 2.12,

f = f ,

and thus

fEhl(OD).

We now show that i i i ) ~ i v ) in Theorem 2.6.

For

f~LI(OD), D x S s ( X )

has a nontangential limit at almost every point

1 N j . f ( P ) + R ~ f ( P ) .

It now follows that if iii) is

QE~D,

and this limit equals -~

satisfied, and

fELI(OD),

then iv) holds.

Conversely, now assume iv). From what we just noted in the above paragraph,

IV Syl6 LI(OD),

and

IIIV S flllLl~ao)<=c{l[ fllLl~ao) + ~7=t llge fllLl(am}.

For any

fEL(cgD),*

it was remarked in [7], that (VSy)* belongs to weak L a of 0D. I n particular, for each r, 0 < r < l , (VSf) ~ belongs to

Lalr+~(OD),

e > 0 . However, by Lemma 2.1, there exists r, 0 < r < l , such that

Ivaf(x)l"

is subharmonic in D. If we choose e > 0 such that l - e > r , and let P = ~ - ' e e ' then p > l , 1 and so we can apply Lemma2.2, to show that

IVSI(X)I'_~P(IVSF)(X),

and so

(VSI)(Q)~_P(IVSF)(Q) 1/'.

Hence we conclude that

f0o (vs,)*

^{dO <=}

c f al, lVS,,(O)ldQ,

and so iv)-~iii). Thus the proof o f Theorem 2.6 is finished.

We give an application of Theorem 2.6 to a special two-dimensional situation.

Theorem 2.13.

Let F be a simple closed C ~ Jordan curve in the complex plane C. Set

1 / . f ( 0 dill

_2Lf

f ( 0 dlCl

Cf(z) = 2-'~i a r TZ-~-~ = !im 2rti' I~-r z - r

where

d[r

are length and z<F, and let H (F) = {fEL x*

(F):

f real valued, fr fd]r

0,

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and CfEZ l(F)}

with

}]fl]ul(r) = Hfi[L 1 (r) + [t CflILl(r).

Then H x (F)*

= B M O (F).

Proof.

We need only observe that

CfELa(F)

if and only if the principal value operators

1 ~ (Ira ( z ) - Im (~)) d I~1

~ L f ( 0 (Re(~z)-r and

T~-frf()

^iz_r

belong to

LI(F).

These two operators are exactly the Riesz operators defined in Theorem 2.6 for t h e domain D = i n s i d e of F. The conclusion of Theorem2.13 is now immediate from Theorems 2.6 and 2.9.

This completes, in the case of bounded C 1 domains in the complex plane, a remark made in [14]. We can conclude, using the notation of [14], that if f/co and C(f/og) are in

LI(A, dl~l),

then f is the real part of the boundary values of an analytic function FEHI(0, dog). The opposite implication was shown in [14].

Some concluding remarks:

Let D be a bounded C 1 domain in R". Let X* be a fixed point in D, and let

dog=do9 x*

be harmonic measure in D, with pole at X*, and

og=-'da

de) be its density with respect to surface measure. I f n = 2 , the equivalence between i) and ii) in Theorem 2.6 follows combining the results in [13] and [14], even in the case of Lipschitz domains. Moreover, if we let

H~OD,

^dog)=

{f, f=u]o o, Au=O

in D and

uEL~(OD,*

do))} (here N stands for 'Dirichlet data'), then, for n--2 and arbitrary Lipschitz domains,

fEH~(OD, dog)

^iff

g = f . ogEHI(OD),

where this is the space defined in this paper. This also follows from [13] & [14].

if again n = 2 , and D is C 1 " [to insure that

daEA~,|

" and we let Also,

k ^o9 )

H I ( D , - ~ ) = { f f ,

~ = V U , VU=O, and

(~)ELI(OD,--~)},*

,,, (oo, -7) --

^{- -}

3--~ ' ~t = V U , a E H 1 (o

and

HIOD, de)= {f, f=ulaD, Au

= 0 in D}, and

uED(OD, da),*

then using [13] and

/ 1 x ,

[14] we can see that

fEH~OD, da)

iff

g=f.ogEHllOD, _~a].

^Moreover,

1 1 N . \ t O , /

de

^{- i}

fa# =0. AJso, fEHi(OD, da )

iff

f=Y~2,b,,

Z ] 2 , 1 < + ~ o , supp

bjcBg

]]b./[[~ a ( B j ) ' 1

fbjdo~=O.

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On the Hardy space H 1 of a C ~ domain 21

W e c o n j e c t u r e t h a t these results also h o l d f o r n > 2 , a n d a l r e a d y have some p a r t i a l positive results i n this direction.

F i n a l l y , we r e m a r k t h a t the analysis o f H P ( O D ) , p < 1, a l o n g the lines o f this paper, o n a b o u n d e d C 1 d o m a i n r e m a i n s open. M o r e o v e r , the study o f the situa- t i o n c o n s i d e r e d here i n the setting o f a n a r b i t r a r y Lipschitz d o m a i n i n R n, n > 2 , is also open, even i n the case p > 1.

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22 Eugene B. Fabes and Carlos E. Kenig: On the Hardy space H x of a C 1 domain

19: STEIN, E. M. and WEIss, G., Introduction to Fourier Analysis on Euclidean spaces, Princeton, NJ, 1971.

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Received September 1, 1979 Eugene B. Fabes

School of Mathematics University of Minnesota Minneapolis

Minnesota 55455 USA and

Carlos E. Kenig

Department of Mathematics Princeton University Princeton

New Jersey 08540 USA

On the Hardy space H 1 o f a C 1 d o m a i n